How do you get shine tv app to access local network so it can chrome cast

It would seem that Shine TV does not support Chromecast through mobile devices at this time

From https://www.shinetv.co.nz/help/articles/watching-chromecast-desktop:

At this time, we do not support Chromecast devices that some TVs have built-in. While it may work, our player is not developed to support this and we cannot troubleshoot any issues that may arise. Only external Chromecast devices are officially supported. While there is no comprehensive list of supported and unsupported models, if you provide the model number to our Support team, we are happy to check for you.

Additionally, it is not possible to Chromecast via a mobile web browser (including Chrome) on iOS.

As long as the iPhone is connected to the local network all Apps on it will use the local network for internet access and device detection.

Does the ShineTV App support ChromeCast? It may only support Airplay.

That would be a question for Google since Chromecast is a Google device.